Exercise 4.9 Consider the matrices Aa, for varying a € R: -1-2a) -1 a +1 1-a 0 1 α 1+ a a) Find the values of a R such that A is diagonalisable over R and write the matrix HEM (3 x 3, R) of the eigenvectors.

Q: Prove that if matrix A is diagonalizable with n real eigenvalues λ1, λ2, . . . , λn, then ∣A∣ = λ1λ2…

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Q: Let A be a 3 × 3 diagonalizable matrix whose eigenvalues are λ₁ = 4, λ2 = −1, and №3 = −2. If V₁ =…

A: We will use diagonalization of matrix and its properties to find required vector

Q: Let A be the 2 x 2 matrix a11 a12 a21 a22 where a11, a12, a21, 922 ER. Show that if a12021 > 0, then…

A: Consider the 2×2 matrix A=a11a12a21a22. Here a11, a12, a21, a22∈ℝ. Assume that a12a21>0. To show…

Q: 1) let A be a 3x3 matrix. A = 0 1 01 1 a) Find the characteristic equation 1 O (b) Find the…

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Q: The value2=0 is an eigenvalue of the matrix A if and only if A is a) nonsingular b) Symmetric c)…

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Q: Let A be and B be n x n matrices, assume that is an eigenvector of the matrix product AB. Assume…

A: be  matrices. is an eigen vector of the matrix

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Q: Let A be an n × n matrix. Prove that A and At have the same eigenvalue equation

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Q: The 2×2 real symmetric matrix A has two distinct eigenvalues λ1 and λ2.If v1 =(1,2) is an…

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Q: Let A be an n× n matrix. Show that. a) If A’ = 0, then zero is the only eigenvalue of A. b) The…

A: The given details: A be an n×n matrix. Take the matrix, A=2314 To show that: a) If A3=0, then zero…

Q: Let A be a 2 x 2 matrix. If det(A) = 3 and tr(A) = -4, the eigen values of A are: A) 2 = 1,3 C) 2 =…

A: Solve by matrix

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Q: Let V₁ = 3 B-B-B and V3 = 0 Av = 3 = be eigenvectors of the matrix A which correspond to the…

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Q: 6 Let b₁=[2] b₂ = 3 4 [ C₁ = [ ² ] ² ₂ = 1²²] > V = [] (2 13 2 and C = { C₁₂ (₂} Find and consider…

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Q: Consider the following matrix: A = The following vectors are linearly independent eigenvectors of A:…

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Q: Exercise 2. Suppose that A and B are matrices of the same size. Prove the following: (a) (aA)¹ = aAª…

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Q: Consider the matrix A = /a11 0 0 a21 a22 0 a31 a32 a334 What is tr(AA')? ] Show that |A| = [1 aii.…

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Q: Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are 11 = 2, 12 = -1, and 13 = -4. If V1 =…

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Q: Let A be a 3×3 symmetric matrix. Assume that A has two eigenvalues: A1 = 0, and X2 = 2. The vectors…

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Q: Consider the matrix A = /a11 0 0 a21 a22 0 a31 a32 a33 What is tr(AA')? ] Show that |A| = [1 aii.…

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Q: Let A be a symmetric n x n matrix. a) Show that if u, ER", then (Au) vu (Av). b) Show that if A has…

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Q: -2 3.

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Q: Let A be a 2 x 2 matrix such that A? = A. Use an example other than identity matrix to show that 1 =…

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Q: 2 3 1 2 and B = 2 3 0 5 [1 2-2: Let A = |1 1 |2 1 3 a) Find A-B", if exists. b) Find the matrix C…

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Q: Find three linearly independent eigenvectors for this matrix P (projection matrices have A1 and A=…

A: Step 1: Step 2: Step 3: Step 4:

Q: For each matrix A, calculate an orthonormal basis of eigenvectors and use it to diagonalize A. [o 1…

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Q: Let A be a 3×3 symmetric matrix. Assume that A has three eigenvalues: A1 = -1, A2 = 2, and A3 = 5.…

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Q: Let A be an invertible, n x n matrix such that A² = A. a) Calculate det(A). b) If n = 3, what is A?…

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